Engineering - Rotational dynamics Flashcards
Definition of Inertia
A mass’ resistance to a change in velocity when subjected to a force
Larger inertia of mass =
larger force required to change its velocity by a certain amount
Definition of moment of inertia
a measure of an objects resistance to being rotationally accelerated about an axis
Equation for the moment of inertia of a point mass
I = mr^+
Equation for the moment of inertia of a point mass
I = mr^2
What does ‘r’ stand for in the equation :
I = mr^2
The distance from the axis of rotation
What is an object made up of more than one point mass called?
An extended object
Equation of the moment of inertia of an object made up of more than one point mass (extended object)
I = ∑mr^2
Factors affecting an object’s moment of intertia
-The object’s total mass
-How its mass is distributed about the axis of rotation, which varies as the distance from the axis of rotation ‘r’ is varied
Equation to find the new moment of inertia ‘Inew’
Inew = I +mr^2
Total KE of an object
The sum of the KEs of all the individual particles making up the objects
Equation for KE linking moment of Inertia, I, and angular speed, ω.
1/2Iω^2
Angular displacement
-θ
-angle through in any given direction in radians
Angular speed
-ω
-angle an object moves through per unit time (only has magnitude)
-measured in rads^-1
Angular velocity
-ω
-angle an object moves through per unit time (has magnitude and direction, clockwise or anticlockwise)
-measured in rads^-1
Angular acceleration
-⍺
-change in angular velocity over time taken
-measured in rads^-2
angular velocity =
angular displacement / time
angular acceleration
angular velocity / time
What does the area of an angular acceleration -time graph show?
The angular velocity
What does the area of an angular velocity-time graph show?
The angular displacement
What is torque?
The product of a force and its distance from the axis of rotation, in units Nm
What are the equations for torque? (two)
-T = Fr
-T = I⍺
What does the law of conservation of angular momentum state?
That when no external torque acts, the angular momentum of the system stats constant.
Equation for conservation of momentum?
I1ω1 + I2ω2 = (I1 +I2)ω
What is angular impulse?
The product of torque and its duration, where the applied torque is constant, and is equal to the change in angular momentum.
Angular Impulse =
TΔt = Δ(Iω)
How do you find the change in angular momentum from a Torque-time graph?
You find the area under the curve
What is work done?
The force causing the motion multiplied by the distance travelled
Work done =
Tθ
What must work done overcome to increase the rotational kinetic energy of an object?
The friction torque
How do you calculate the friction torque experimentally?
- Apply an accelerating torque to the wheel to bring it up to a certain velocity
- Remove the accelerating torque and measure the time taken for the wheel to come to rest
- Calculate average deceleration using :
⍺ = (ω2 - ω1) / t - Calculate frictional torque using T = I⍺
( assuming that the moment of inertia is known)
How do you work out work done from a Torque-angular displacement graph?
The area under the curve
